Some non-equilibrium invariant states for the asymmetric exclusion QMS at level one

We review the Asymmetric Exclusion QMS in the light of new results, taking as a starting point the dynamics in the one particle space. We give a condition for the Asymmetric Exclusion QMS to be conservative, prove that an invariant state is necessarily diagonal and give conditions on eigenvalues of such an invariant state. We also give, a necessary condition to annul the generator of the predual semigroup; with this and the weighted detailed balance condition, we propose a method to construct some non-equilibrium invariant states.